July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a essential concept that pupils are required understand due to the fact that it becomes more important as you advance to more complex math.

If you see higher mathematics, such as differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you hours in understanding these ideas.

This article will talk about what interval notation is, what are its uses, and how you can decipher it.

What Is Interval Notation?

The interval notation is merely a way to express a subset of all real numbers through the number line.

An interval refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)

Basic problems you face essentially consists of one positive or negative numbers, so it can be difficult to see the utility of the interval notation from such simple applications.

Though, intervals are generally employed to denote domains and ranges of functions in higher math. Expressing these intervals can increasingly become complicated as the functions become progressively more complex.

Let’s take a straightforward compound inequality notation as an example.

  • x is greater than negative 4 but less than 2

Up till now we know, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, it can also be denoted with interval notation (-4, 2), denoted by values a and b separated by a comma.

So far we know, interval notation is a method of writing intervals concisely and elegantly, using predetermined rules that make writing and understanding intervals on the number line simpler.

In the following section we will discuss about the principles of expressing a subset in a set of all real numbers with interval notation.

Types of Intervals

Several types of intervals place the base for writing the interval notation. These kinds of interval are essential to get to know because they underpin the entire notation process.


Open intervals are used when the expression does not include the endpoints of the interval. The prior notation is a good example of this.

The inequality notation {x | -4 < x < 2} express x as being greater than -4 but less than 2, which means that it excludes neither of the two numbers mentioned. As such, this is an open interval denoted with parentheses or a round bracket, such as the following.

(-4, 2)

This represent that in a given set of real numbers, such as the interval between negative four and two, those two values are not included.

On the number line, an unshaded circle denotes an open value.


A closed interval is the opposite of the last type of interval. Where the open interval does not include the values mentioned, a closed interval does. In text form, a closed interval is written as any value “greater than or equal to” or “less than or equal to.”

For example, if the last example was a closed interval, it would read, “x is greater than or equal to -4 and less than or equal to 2.”

In an inequality notation, this can be expressed as {x | -4 < x < 2}.

In an interval notation, this is stated with brackets, or [-4, 2]. This means that the interval includes those two boundary values: -4 and 2.

On the number line, a shaded circle is employed to represent an included open value.


A half-open interval is a blend of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the prior example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than two.” This implies that x could be the value -4 but couldn’t possibly be equal to the value 2.

In an inequality notation, this would be written as {x | -4 < x < 2}.

A half-open interval notation is denoted with both a bracket and a parenthesis, or [-4, 2).

On the number line, the shaded circle denotes the number present in the interval, and the unshaded circle denotes the value excluded from the subset.

Symbols for Interval Notation and Types of Intervals

In brief, there are different types of interval notations; open, closed, and half-open. An open interval excludes the endpoints on the real number line, while a closed interval does. A half-open interval includes one value on the line but does not include the other value.

As seen in the prior example, there are various symbols for these types subjected to interval notation.

These symbols build the actual interval notation you develop when expressing points on a number line.

  • ( ): The parentheses are used when the interval is open, or when the two endpoints on the number line are not included in the subset.

  • [ ]: The square brackets are utilized when the interval is closed, or when the two points on the number line are not excluded in the subset of real numbers.

  • ( ]: Both the parenthesis and the square bracket are used when the interval is half-open, or when only the left endpoint is excluded in the set, and the right endpoint is included. Also known as a left open interval.

  • [ ): This is also a half-open notation when there are both included and excluded values among the two. In this instance, the left endpoint is included in the set, while the right endpoint is excluded. This is also called a right-open interval.

Number Line Representations for the Various Interval Types

Aside from being written with symbols, the different interval types can also be represented in the number line employing both shaded and open circles, depending on the interval type.

The table below will display all the different types of intervals as they are represented in the number line.

Interval Notation


Interval Type

(a, b)

{x | a < x < b}


[a, b]

{x | a ≤ x ≤ b}


[a, ∞)

{x | x ≥ a}


(a, ∞)

{x | x > a}


(-∞, a)

{x | x < a}


(-∞, a]

{x | x ≤ a}


Practice Examples for Interval Notation

Now that you know everything you are required to know about writing things in interval notations, you’re prepared for a few practice problems and their accompanying solution set.

Example 1

Convert the following inequality into an interval notation: {x | -6 < x < 9}

This sample question is a easy conversion; just use the equivalent symbols when denoting the inequality into an interval notation.

In this inequality, the a-value (-6) is an open interval, while the b value (9) is a closed one. Thus, it’s going to be expressed as (-6, 9].

Example 2

For a school to participate in a debate competition, they need at least three teams. Represent this equation in interval notation.

In this word question, let x be the minimum number of teams.

Because the number of teams required is “three and above,” the value 3 is included on the set, which states that three is a closed value.

Additionally, since no maximum number was stated regarding the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.

Therefore, the interval notation should be denoted as [3, ∞).

These types of intervals, when one side of the interval that stretches to either positive or negative infinity, are also known as unbounded intervals.

Example 3

A friend wants to undertake a diet program constraining their regular calorie intake. For the diet to be successful, they must have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you write this range in interval notation?

In this word problem, the number 1800 is the lowest while the value 2000 is the highest value.

The problem implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.

Therefore, the interval notation is described as [1800, 2000].

When the subset of real numbers is restricted to a range between two values, and doesn’t stretch to either positive or negative infinity, it is also known as a bounded interval.

Interval Notation Frequently Asked Questions

How Do You Graph an Interval Notation?

An interval notation is basically a technique of describing inequalities on the number line.

There are laws of expressing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is denoted with an unshaded circle. This way, you can promptly see on a number line if the point is excluded or included from the interval.

How To Change Inequality to Interval Notation?

An interval notation is basically a different technique of expressing an inequality or a combination of real numbers.

If x is higher than or lower than a value (not equal to), then the number should be written with parentheses () in the notation.

If x is higher than or equal to, or less than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation prior to check how these symbols are utilized.

How To Rule Out Numbers in Interval Notation?

Values excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the value is ruled out from the combination.

Grade Potential Can Help You Get a Grip on Mathematics

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